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This got resolved, so here's what happened for anyone else with this issue.

After 6 weeks of the page ranking well on mobile, but not ranking at all on desktop, we:

• specified the old URL in the sitemap.xml
• put in an individual 301 from the old URL to the new (ie didn't just rely on the rules we were using for other URLs)
• ensured only one redirect was required from old to new URL
• put a link on the homepage, using the targeted keyword phrase, to the new URL
  (these were in addition to the other technical fixes detailed in the original post above)

About a week after this, the desktop ranking returned, with the new URL ranking.  Maybe coincidence, maybe the extra steps were required. But it's worked, whatever the reason.

Interesting points, thanks. The domain hasn't changed, only the URL, so I've manually updated the sitemap to add in an old / problem URL and we'll see if Google restores the relationship between old and new URL. Will update the thread with any changes.

Thanks.

A client's developer moved a site onto a new (WordPress) CMS, where the only change was URLs - the front end code stayed the same. The site is 10+ years old and previously had fantastic rankings (#1-4) with inner pages for some relatively generic search phrases (eg 10,000 searches / month in the UK, per Keyword Planner).

Now, on Desktop searches the site isn't appearing anywhere in the 300+ results for a key search phrase, where it used to rank between #2-4; however over the last 3 weeks on Mobile the site ranks better than before, even though the site isn't at all mobile-friendly (it's over 10 years old).

During the move, there were some errors by their developer:

• mistakenly left in a sitewide rel=canonical tag referring to the homepage
• 3-4 301s before finally reaching new URLs
• a lot of 301s missed (250+ crawl errors appeared in Search Console)
• page content differentiation by parameter, instead of individual URLs

For example, the page that used to rank for the targeted phrase, this left 4 different URLs indexed, with the same content.

To tackle this, we have so far:

• put in correct rel=canonical tags
• set up Search Console to recognise URL parameter as differentiating content
• fixed all crawl errors appearing in Search Console
• added a link direct to the problem page, direct from the homepage
• stopped duplicate content being indexed (including for the page in question)
• ensured the page load speed is still good (< 0.75s)

Ranking for Desktop over Mobile would make sense, but not Mobile over Desktop!  I'd really appreciate any advice on how to tackle this.

Thanks!